Before going to the distribution of
\(((T_j)_{j:T_j\le t},N(t))\) , we give a more general result of
Theorem REF , the proof of which gives an alternative proof for
the theorem, namely we derive the corresponding result for the time
changed Lévy processes: Cox processes are obtained by selecting
the standard Poisson for the original Lévy process.
Note that the time changed Lévy processes by the
subordinators are shown to be Lévy processes again (see e.g. [1]} and [2]}). We show a kind of
reverse relation which seems to be new.
